Math · Unit 17

Calculus

Limits, derivatives, integrals and their applications

Calculus handles two questions that resisted mathematics for centuries: the exact rate of change at an instant, and the exact area under a curve. Both are solved by the same limiting idea.

The unit covers limits, derivatives and their applications, integrals, and areas - ending with the fundamental theorem that ties differentiation and integration together as inverses.

This unit breaks down into 21 short steps and 120 questions, starting at difficulty 4 and building to 5. Below you can see exactly what it covers, how the path is structured, and worked examples with explanations.

Steps: 21
Questions: 120
Difficulty: 4-5

What this unit covers

Using Derivatives
Integrals
Derivatives
Areas and Applications
Limits

Where this fits

Needs Precalculus. The most demanding unit in the Math track.

Where people slip

A derivative is a rate, not a value. Confusing the value of a function with the value of its derivative at the same point is the error that recurs most in the unit.

How the unit is structured

Calculus runs as 21 short steps that unlock in order. 15 are practice rounds and 6 are challenge rounds that pull together everything before them. Questions start at difficulty 4 and climb to 5 as you progress.

Step 1 · easierStep 21 · harder

Challenge rounds

Example questions

30 real questions from this unit, with the answer and the reason behind it, grouped by what they practise. There are 120 in the unit altogether.

Areas and Applications

Build the sentenceLevel 4
1. Build a true sentence about what an integral finds.
Answer: An integral finds the area under a curve
A definite integral computes the area under a curve.
Fact or fibLevel 4
2. The definite integral of velocity over a time interval gives the displacement.
Answer: True
Integrating velocity accumulates position change, which is displacement.
Fill the blankLevel 4
3. The area under a curve is found using a ____ integral.
- definitecorrect
- indefinite
- partial
- double
A definite integral has limits and produces a numerical area.
Choose all that applyLevel 5
4. Select all correct statements about a definite integral.
- It can represent the area under a curvecorrect
- It gives a number, not a function
- It always needs + C
- It equals F(b) - F(a) by the FTCcorrect
A definite integral gives a number (an area or accumulation) equal to F(b) - F(a); it does not carry a + C.
Guess the numberLevel 5
5. What is the area under y = 3x^2 from x = 0 to x = 2? (the antiderivative of 3x^2 is x^3)
Answer: 8
Evaluate x^3 from 0 to 2: 2^3 - 0 = 8.
Match the pairsLevel 5
6. Match each quantity to the integral that produces it.
Answer: Displacement = integral of velocity; Area under a curve = integral of f(x); Average value = integral of f divided by (b - a)
Displacement comes from integrating velocity, area from integrating f(x), and the average value divides the integral by the interval width.

Derivatives

Build the sentenceLevel 4
7. Build a true sentence about what a derivative measures.
Answer: The derivative measures the rate of change
The derivative measures how fast a quantity changes, its rate of change.
Choose all that applyLevel 4
8. Select all TRUE derivative statements.
- d/dx of x^2 = 2xcorrect
- d/dx of x^3 = 3x^2correct
- d/dx of x = 0
- d/dx of 5 = 5
d/dx of x^2 = 2x and d/dx of x^3 = 3x^2 are correct; d/dx of x = 1 and d/dx of 5 = 0.
Fact or fibLevel 4
9. The derivative of sin(x) is cos(x).
Answer: True
d/dx of sin(x) = cos(x) is one of the basic trigonometric derivatives.
Fill the blankLevel 4
10. d/dx of x^4 = ____.
- 4x^3correct
- 4x^2
- x^3
- 3x^4
The power rule gives 4 times x^3 = 4x^3.
Match the pairsLevel 4
11. Match each function to its derivative.
Answer: x^2 = 2x; x^3 = 3x^2; x^4 = 4x^3
The power rule takes x^n to n times x to the (n - 1) power.
Multiple choiceLevel 4
12. Using the power rule, what is d/dx of x^2?
- 2xcorrect
- x
- 2
- x^2
The power rule brings the exponent down: d/dx of x^2 = 2x.

Integrals

Choose all that applyLevel 4
13. Select all correct antiderivatives (indefinite integrals).
- integral of 1 dx = x + Ccorrect
- integral of 0 dx = Ccorrect
- integral of 2x dx = 2 + C
- integral of x dx = x + C
The integral of 1 dx = x + C and the integral of 0 dx = C; the other two are wrong.
Fill the blankLevel 4
14. The process of finding an antiderivative is called ____.
- integrationcorrect
- differentiation
- factoring
- substitution
Integration is the reverse of differentiation and finds antiderivatives.
Match the pairsLevel 4
15. Match each function to its antiderivative (ignore + C).
Answer: 2x = x^2; 3x^2 = x^3; 1 = x
Reverse the power rule: integrating 2x gives x^2, 3x^2 gives x^3, and 1 gives x.
Multiple choiceLevel 4
16. What is the integral of 2x dx?
- x^2 + Ccorrect
- 2 + C
- x^2
- 2x^2 + C
The antiderivative of 2x is x^2, plus the constant of integration C.
Odd one outLevel 4
17. Which of these is NOT an antiderivative of 2x?
- 2x^2correct
- x^2
- x^2 + 1
- x^2 - 3
Antiderivatives of 2x all look like x^2 + C; 2x^2 is not one, since its derivative is 4x.
Sort into groupsLevel 5
18. Sort each description as a Definite or Indefinite integral.
Answer: integral of 2x dx from 0 to 3 = Definite; integral of 2x dx with no limits = Indefinite; gives a single number = Definite; gives a family of functions + C = Indefinite
A definite integral has limits and gives a number; an indefinite one gives a family of functions + C.

Limits

Build the sentenceLevel 4
19. Build a true sentence about what a limit describes.
Answer: A limit describes a function near a point
A limit tells you the value a function approaches near a point.
Fact or fibLevel 4
20. As x approaches 0 from the right, 1/x grows without bound toward infinity.
Answer: True
Dividing 1 by a tiny positive number gives a huge result, so 1/x heads to infinity.
Fill the blankLevel 4
21. If the limit of f(x) as x approaches a equals f(a), then f is ____ at a.
- continuouscorrect
- increasing
- undefined
- constant
That matching condition is exactly the definition of continuity at a point.
Multiple choiceLevel 4
22. What is the limit of (x^2 - 1)/(x - 1) as x approaches 1?
- 2correct
- 0
- 1
- undefined
Factor the top as (x - 1)(x + 1) and cancel; you are left with x + 1, which is 2 at x = 1.
Odd one outLevel 4
23. Which of these is NOT an indeterminate form?
- 5 / 1correct
- 0 / 0
- infinity / infinity
- infinity - infinity
5 / 1 simply equals 5, so it is determinate, unlike the classic indeterminate forms.
Choose all that applyLevel 5
24. Select all limits that equal 0.
- limit of 1/x as x approaches infinitycorrect
- limit of 1/x^2 as x approaches infinitycorrect
- limit of (x + 1) as x approaches 0
- limit of x/x as x approaches 5
1/x and 1/x^2 both go to 0 at infinity; the other two limits equal 1.

Using Derivatives

Build the sentenceLevel 4
25. Build a true sentence about the derivative at a smooth maximum.
Answer: At a maximum the derivative equals zero
At an interior maximum of a smooth curve, the tangent is flat, so the derivative equals zero.
Fact or fibLevel 4
26. If f'(x) > 0 on an interval, then f is increasing on that interval.
Answer: True
A positive derivative means the slope is upward, so the function rises.
Fill the blankLevel 4
27. A point where f'(x) = 0 is called a ____ point.
- criticalcorrect
- inflection
- endpoint
- boundary
Where the derivative is zero, the tangent is flat, marking a critical point.
Guess the numberLevel 4
28. The slope of the tangent to y = x^2 at x = 4 is f'(4). What number is it?
Answer: 8
Since f'(x) = 2x, the slope at x = 4 is 2 times 4 = 8.
Multiple choiceLevel 4
29. The line tangent to y = x^2 at x = 3 has what slope? (recall f'(x) = 2x)
- 6correct
- 9
- 3
- 2
Since f'(x) = 2x, the slope at x = 3 is 2 times 3, which is 6.
Choose all that applyLevel 5
30. Select all that indicate a local minimum of a smooth function at x = c.
- f'(c) = 0 and f''(c) > 0correct
- f' changes from negative to positive at ccorrect
- the graph has its lowest nearby value at ccorrect
- f''(c) < 0
A local min shows f'(c) = 0 with f''(c) > 0, f' changing negative to positive, and the lowest nearby value; f''(c) < 0 would mean a maximum.

Where these questions come from. Each unit starts as a plan of the concepts it should cover and the difficulty it should span. Questions are written against that plan with AI assistance, then checked by a validator that rejects anything without a single defensible answer, an explanation, or plausible wrong options. How we write questions sets out the whole process, and corrections are fixed in the bank and reach the site and the app the same day.

How you practise

This unit mixes 14 different question formats, so you are recalling and applying rather than recognising the same layout every time.

Build the sentence
Choose all that apply
Fact or fib
Fill the blank
Guess the number
Match the pairs
Multiple choice
Odd one out
Picture question
Put in order
Sort into groups
Spell it
True or false
Type the answer

Practise Calculus

120 questions across 21 steps. Start with step one and crawl at your own pace.

Play this unit

Read about Calculus

Explainers from our blog on what this unit covers. Each one ends with real questions from the bank.

What Is Calculus? The Mathematics of Change and AccumulationCalculus is the mathematics of things that change continuously: speeds, slopes, areas and totals. What its two halves do, and why they are one subject.September 14, 2026 · 5 min read

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